Long-Run Covariability

Ulrich K. Müller and Mark W. WatsonPrinceton University

September 2017

Motivation

• Study the long-run covariability/relationship between economic variables

⇒ great ratios, long-run Phillips curve, nominal exchange rates and relativeprice levels, etc.

• Challenge: many economic time series are persistent

⇒ spurious regression effects

• Cointegration framework is highly constraining

⇒ Very specific model of persistence, rigid relationship between persistenceand long-run covariability

1

This Paper

• Defines population measures about long-run covariability of general bivari-ate process

• Derives confidence intervals about these measures that are valid in flexibleparametric model of long-run properties

⇒ “Descriptive statistics with confidence intervals”

• Statistical framework reflects sparsity of long-run information (cf. Müllerand Watson (2008, 2016a, 2016b))

⇒ No consistent estimation of long-run properties

⇒ Inference in presence of nuisance parameters (Elliott, Müller and Watson(2015), Müller and Norets (2016))

2

Long-Run Correlation Matrix

× TFP 091∗ 053∗ 098∗ 078∗

90% CI [071; 097] [002; 081] [095; 099] [045; 095]

053∗ 092∗ 070∗

90% CI [003; 081] [068; 097] [028; 091]

051∗ 03890% CI [002; 080] [−008; 071]

× 072∗

90% CI [038; 093]

3

Outline of Talk

1. Introduction

2. Statistical framework, running examples

3. Measuring long-run covariability

4. A flexible parametric model of long-run properties

5. Construction of confidence intervals

6. Additional applications

4

GDP and Consumption Growth

5

GDP and Consumption Growth

6

Low-Pass Moving Averages vs Long-Run

Projections

7

Cosine Transforms and Long-Run Projections

• Let Ψ() =√2 cos()

• For a scalar sequence =1, define

=1√

X=1

Ψ( )

=

√2√

X=1

cos( ), = 1 · · ·

8

= 12 Cosine Weights

9

GDP Growth

10

= 12 Cosine Transforms for GDP

11

Long-Run Projections

• Define as the predicted values from a regression of on Ψ( )=1

• Equivalently

=1√

X=1

Ψ( ) =

√2√

X=1

cos( )

because the weights Ψ( ) are orthonormal

• usefully thought of as low-pass filter for frequencies lower than 2periods (66 years of data, = 12 ⇒ lower than 11 year cycles)

• fully characterized by regression coefficients

12

GDP and Consumption Growth

13

Short and Long-Term Interest Rates

14

Asymptotic Behavior of Cosine Transforms

• Let ( 0

0 )0 be 2 × 1 cosine transforms of ( )=1

• Parameters of interest and suggested confidence intervals are defined interms of (0

0 )0

• Consider asymptotics where is fixed, as in Müller (2004, 2007, 2014),Phillips (2006), Müller and Watson (2008, 2013, 2016a, 2016b), Sun(2013, 2016)

⇒ Defines notion of “long-run”: Periods of interest are 2 and longer

⇒ Reflects that in samples of interest, reasonable are small

⇒ Ignoring data information beyond (0

0 )0 avoids modelling higher

frequency properties

15

Asymptotic Behavior of Cosine Transforms

• Assume (∆∆) stationary with a spectral density . Let () = ()|1− i|2 be the pseudo-spectrum of ( ), and suppose

( )→ ()

in a suitable sense.

• Under linear process assumption and additional regularity conditions, byCLT in Müller and Watson (2016)Ã

!⇒

Ã

!∼ N (0Σ) , Σ =

ÃΣ ΣΣ Σ

!and

Σ = Var

Ã

!→ Σ

where Σ is a function of .16

Measuring Long-Run Covariability

Ω = −1X=1

⎡⎣Ã

!Ã

!0⎤⎦ = X=1

⎡⎣Ã

!Ã

!0⎤⎦=

ÃtrΣ trΣtrΣ trΣ

!→

ÃtrΣ trΣtrΣ trΣ

!

=

ÃΩ ΩΩ Ω

!= Ω

• Scalar parameters derived from Ω

— = Ω √ΩΩ

— = Ω Ω

— =qΩ −Ω2 Ω

17

GDP and Consumption Growth

18

Interpretation of and

• Define

= argmin

⎡⎣−1 X=1

( − )2

⎤⎦ = argmin

⎡⎣ X=1

( − )2

⎤⎦ =

vuuutmin

⎡⎣−1 X=1

( − )2

⎤⎦ =vuuutmin

⎡⎣ X=1

( − )2

⎤⎦

• Then → and → , since

⎡⎣−1 X=1

( − )2

⎤⎦ = −1

⎡⎣ X=1

2 − 2X=1

+ 2X=1

2

⎤⎦= Ω − 2Ω + 2Ω

• Population 2 is Ω2(ΩΩ )→ 2

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Comparison with Standard Spectral Analysis

• Recall that is pseudo-spectrum of ( ), and ( )→ ()

• Up to reasonable approximation, Ω is to average over (pseudo) spec-tral density () over frequencies ∈ [ ], corresponding toaverage of () over ∈ [ ].

• Two departures from classic spectral analysis:

1. Derive inference for fixed, no LLN applicable

2. Allow for curvature in that is non-negligble in 1 neighborhood(=don’t assume is flat).

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Parametrizing , and Implied Ω

Two important special cases:

1. In I(0) model, () ∝ Λ, ( )0 ∼ N (0Λ)

⇒ Ω ∝ Λ

2. In I(1) model, () ∝ Λ2, ( )0 ∼ N (0Λ2)

⇒ Ω ∝ Λ

21

Short and Long-Term Interest Rates

22

I(0) and I(1) Inference

⇒ Follows from well-known small sample results (cf. Anderson (1984))

and I-RatesI(0) I(1) I(0) I(1)

Estimates 0.93∗ 0.93∗ 0.97∗ 0.94∗

90% CI [0.81;0.97] [0.82;0.97] [0.93;0.99] [0.82;0.97]

Estimates 0.76∗ 0.84∗ 0.96∗ 0.85∗

90% CI [0.60;0.92] [0.67;1.01] [0.84;1.08] [0.68;1.03]

Estimates 0.35 0.35 0.63 0.4890% CI [0.26;0.55] [0.26;0.54] [0.47;0.97] [0.36;0.74]

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A Flexible Parametric Model for

• ( ) model:

() =

Ã(2 + 21)

−1 0

0 (2 + 22)−2

!0 +0

with ∈ [−04 1], ∈ [0∞] and lower triangular

• 11 parameter model that embeds bivariate fractional, local-to-unity andlocal-level models as special cases

• Allows for various long-run phenomena such as (stochastic) breaks inmeans, slow mean reversion, overdifferencing, etc.

• Ω can be computed from implied Σ = Σ(), = ( )

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Construction of Confidence Intervals

• Under asymptotic approximation, a parametric small sample problem:

— Observe = ( 0 0)0 ∼ N (0Σ), where Σ = Σ(), =

( ) ∈ Θ

— Seek confidence interval () for parameter of interest = (), forknown

— Impose appropriate invariance on

• -weighted average expected length minimizing program

min

Z[lgth(())] () s.t. (() ∈ ()) ≥ 1− ∀ ∈ Θ

⇒ Optimal depends on unknown Lagrange multipliers

25

Form of Optimal Confidence Set

• For simplicity, ignore invariance. Then (cf. Pratt 1961), optimal invertstests 0 of 0 : () = 0 of the form

0() = 1[Z() () cv

Z()Λ0()]

for some probability distribution Λ0 on Θ with support on a subset of : () = 0, Λ0( : (() ∈ ()) 1 − ) = 0 and[0()] ≤ for all () = 0

• Similar form under invariance, but not a family of distributions Λ if para-meter of interest is affected by invariance

⇒ see Müller and Norets (2016) for details. Effective dimension of para-meter space after invariance is 8 for all three problems.

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Computation of Λ

• Arbitrary Λ induces lower bound on expected length criterion that holdsfor all valid CIs (cf. Elliott, Müller and Watson (2015))

• Basic algorithm

1. Let Θ = 1 be candidate set for support of Λ

2. Compute Λ weights and cv such that size of test is − on Θ.

3. Check size control on Θ:

(a) If size is controlled, we are done (compare length to bound generatedfrom cv0 that induces Λ-weighted size equal to ).

(b) If size violated, add violating to Θ and go to Step 2.

27

Computational Details

• Λs have about 30-100 points of support

• CS are within 5% of lower bound on length criterion for = 01

• Size control: 500 consecutive BFGS searches with random starting valuesdon’t find violation

• Single problem takes about 30 minutes on fast PC using Fortran

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Choice of Weighting Function

• Recall

() =

Ã(2 + 21)

−1 0

0 (2 + 22)−2

!0 +0

• Set = 0, 1 = 2 = 0, 1 2 ∼ [−04 14],

∼ (1) diag(150 1) (2)

where ∼ [0 1] and () is 2× 2 rotation matrix of angle .

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Credibility of Resulting CS

• Optimal () could be empty for some , or unreasonably short

⇒ Perversely, this is optimal for very uninformative draws , as coveragethen costly in terms of length

• Generic problem in nonstandard problems: Frequentist (optimality) prop-erties don’t rule out unreasonable descriptions of uncertainty for somerealizations

⇒ Analyzed in detail in Müller and Norets (2016), building on old literature(Fisher (1956), Buehler (1959), Wallace (1959), Cornfield (1969), Pierce(1973), Robinson (1977), etc.)

• Suggested solution: constrained to be superset of equal-tailed credibleset relative to prior

30

Spurious Regression?

• Minimal coverage of weighted average length minimizing nominal 90% CIsfor under = 0

DGP\Assumed Model I(0) I(1) I(0) 0.90 0.01 0.91I(1) 0.01 0.90 0.90 0.01 0.00 0.90

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Results in Running Examples

I(0) I(1) and

Estimates 0.93∗ 0.93∗ 0.91∗

90% CI [0.81;0.97] [0.82;0.97] [0.71;0.97]

Estimates 0.76∗ 0.84∗ 0.76∗

90% CI [0.60;0.92] [0.67;1.01] [0.48;0.95]

Estimates 0.35 0.35 0.4090% CI [0.26;0.55] [0.26;0.54] [0.27;0.66]

I-Rates Estimates 0.97∗ 0.94∗ 0.96∗

90% CI [0.93;0.99] [0.82;0.97] [0.89;0.99]

Estimates 0.96∗ 0.85∗ 0.92∗

90% CI [0.84;1.08] [0.68;1.03] [0.75;1.14]

Estimates 0.63 0.48 0.7090% CI [0.47;0.97] [0.36;0.74] [0.53;0.92]

32

Real Variables: Correlation

× TFP 091∗ 053∗ 098∗ 078∗

90% CI [071; 097] [002; 081] [095; 099] [045; 095]

053∗ 092∗ 070∗

90% CI [003; 081] [068; 097] [028; 091]

051∗ 03890% CI [002; 080] [−008; 071]

× 072∗

90% CI [038; 093]

33

PCE and CPI Inflation

34

PCE and CPI Inflation

Estimate 0.98∗ 1.13∗

90% CI [0.95;0.99] [0.98;1.24]

35

PCE Inflation and Unemployment

36

PCE Inflation and Unemployment

Estimate 0.25 0.2190% CI [-0.27;0.82] [-0.24;0.78]

37

PCE Inflation and Short I-Rates

38

PCE Inflation and Short I-Rates

Estimate 0.47 0.7390% CI [0.00;0.91] [-0.09;1.91]

39

TFP and Unemployment

40

TFP and Unemployment

Estimate -0.65∗ -1.00∗

90% CI [-0.75;-0.35] [-1.64;-0.27]

41

Conclusions

• Study statistical significance of long-run relationships that is robust tomany DGPs

• Underlying derivations are involved, but not difficult or computationallyintensive to apply to data

• In paper: Sensitivity of results to alternative and frequencies of interest

42